CHAPTER X

EXISTENCE THEOREMS

There are of course many lands of existence theorems in analysis
and this chapter only deals with one kind, namely, those which are linked
to the notion of completeness•; roughly speaking, the most intuitive result
(10.1.3) says that when in a Banach space the identity mapping is ** slightly"
perturbed by an additional term in the neighborhood of a point, it still
remains a homeomorphism around that point. The word "slightly" has
to be understood in a precise way, which means more than mere " smallness"
of the perturbing function (see Section 10.2, Problem 2), and has to do with
a limitation on the rate of variation of that function, generally referred to as
a condition of " lipschitzian" type. As a consequence, the natural field of
application of theorems of that type consists of equations in which some
limitation is known on the derivatives of the given functions; furthermore, the
existence theorems obtained in that way are of a local nature. In the next
chapter, we will meet different kinds of existence theorems, which can be
applied to global problems.

The main applications of the general existence theorems of Section 10.1
are; (1) the implicit function theorem (10.2.1) together with its consequence,
the rank theorem (10.3.1) which (locally) reduces to a canonical form the
continuously diiferentiable mappings of constant rank in finite dimensional
spaces; (2) the Cauchy existence theorem for ordinary differential equations
(10.4.5) with its various improvements and consequences; both theorems are
among the most useful tools of both classical and modern analysis. Of course,
what is said of differential equations in this and the next chapter is only a tiny
fraction of that vast theory; other parts of it will be examined in Chapters
XXIII and XXV; the reader who wants to go farther in that direction is
referred to the books of Coddington-Levinson [9], Ince [12] and Kamke [14].

As a last application, we have given a proof of the theorem of Frobenius
(10.9.4), which, as we state it, appears as a natural extension of the Cauchy

264ve in C.